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Comparing options

Ratio, rates and money · level 6, lesson 6 of 7

A question to start

Two phone plans:

  • Plan A: 45 dollars a month, unlimited data.
  • Plan B: 25 dollars a month, plus 4 dollars for every gigabyte (GB) of data you use.

Which is cheaper?

The honest answer is: it depends, and the useful answer is exactly what it depends on. If you use less than 5 GB a month, Plan B is cheaper; if you use more, Plan A is. Big money decisions (phones, cars, homes, bank accounts) almost never have one right answer for everybody. What math gives you is a way to find the right answer for you, and to see through offers that are designed to look better than they are.

The idea

Step 1: put everything on the same footing

To compare options fairly, compare the total cost over the same period, including every cost. This is the money version of comparing unit prices (level 3): you can only compare like with like.

  • Turn monthly costs, yearly costs and one-time costs into one total (for example, over 2 years).
  • Include fees, deposits, taxes and what the thing is worth at the end.
  • Subtract anything you get back (selling the car later, a refund).

Step 2: find the break-even point

When one option has a fixed cost and another a cost per use, there is a usage where they cost the same: the break-even point. Below it one option wins; above it the other does.

For the phone plans, with \(g\) GB a month:

\[\text{Plan A} = 45, \qquad \text{Plan B} = 25 + 4g.\]

They are equal when \(25 + 4g = 45\), so \(4g = 20\) and \(g = 5\).

5 GB 10 GB 45 25 A B B cheaper A cheaper

Why does this work? Plan B costs 4 dollars more for each extra GB, while Plan A stays the same. Plan B starts 20 dollars cheaper, and each GB eats 4 dollars of that head start; after \(20 \div 4 = 5\) GB the head start is gone.

Then look at your own usage, honestly, over a few months, and choose. If you use 3 GB, Plan B costs 37 dollars; if you use 8 GB, it costs 57 dollars.

Paying now or paying over time

Many things can be bought outright or paid off monthly: a phone "included" in a plan, furniture "on easy payments", a car. Always compute the total of the monthly payments and compare it with the price. The difference is the cost of paying over time, which is interest even when nobody calls it that.

Depreciation and appreciation

Things change value after you buy them.

  • Depreciation means losing value. Cars, phones and computers depreciate. A new car often loses a large part of its value in the first few years (the exact amount varies a lot by model).
  • Appreciation means gaining value. A home may appreciate over the years, and so may some collectibles. This is not guaranteed: home prices can also fall, sometimes for years.

Two models of depreciation:

  • Straight-line: the same amount each year. A 1500 dollar laptop losing 300 dollars a year is worth 1200, 900, 600, ... Linear, like simple interest.
  • Percent (declining balance): the same percent each year, so the value is multiplied by the same factor: \(V = P(1 - d)^n\). A car losing 15% a year is multiplied by 0.85 each year. This is compound interest running backwards, and the value never quite reaches zero.

Appreciation at a fixed percent is exactly compound growth: \(V = P(1 + a)^n\).

Buying, leasing and renting

  • Buying: you pay the full price (often with a loan), you own the thing, and at the end you have whatever it is still worth.
  • Leasing (common for cars): you pay monthly to use it for a set time, often with a payment up front, then give it back. Leases often have limits, such as a maximum number of kilometres, with fees for going over. You own nothing at the end.
  • Renting (common for homes): you pay to live there; the owner pays for the building and its repairs and keeps any rise in its value.

To compare buying and leasing, compare the net cost: for buying, the price minus what it is worth when you sell it (plus any loan interest); for leasing, all the payments and fees.

Reading the fine print

Offers are written to look good. Before you sign, look for:

  • Introductory rates: "0% for 6 months" may become 25% after that.
  • Fees: monthly account fees, annual card fees, setup fees, cancellation fees, late fees.
  • Limits: a "free" bank account with only a few free transactions a month, a lease with a kilometre limit.
  • The total: "only 25 dollars a month" means nothing until you know for how many months.

Prices, fees and rates in this lesson are examples; real ones change, so check the current ones.

Worked examples

1. Break-even. Plan A is 45 dollars a month; Plan B is 25 dollars plus 4 dollars per GB.

  • Equal when \(25 + 4g = 45\): \(g = 5\) GB. Use less than 5 GB, choose B; more, choose A.

2. A phone over 2 years. Option 1: buy a phone for 1000 dollars and a plan at 40 dollars a month. Option 2: a plan "with the phone included" at 85 dollars a month for 24 months.

  • Option 1: \(1000 + 24 \times 40 = 1960\) dollars.
  • Option 2: \(24 \times 85 = 2040\) dollars.
  • Option 2 costs 80 dollars more: that is the price of paying for the phone over time.

3. A car depreciating. A 30 000 dollar car loses 15% of its value each year. Its value after 4 years?

  • \(30\,000 \times 0.85^4 \approx\) 15 660.19 dollars. Year by year: 25 500, 21 675, 18 423.75, 15 660.19.

4. Buy or lease? The same car over 4 years. Buying: pay 30 000 dollars, sell for about 15 660 dollars after 4 years. Leasing: 2000 dollars up front plus 400 dollars a month for 48 months.

  • Buying, net: \(30\,000 - 15\,660 = 14\,340\) dollars (plus interest if you borrowed the money).
  • Leasing: \(2000 + 48 \times 400 = 21\,200\) dollars, and you own nothing.
  • In this example buying is cheaper by about 6860 dollars, if you can pay or borrow cheaply. Insurance and repairs would change the picture, so a real comparison includes them too.

5. A home appreciating. If a 400 000 dollar home appreciated by 3% a year for 10 years, its value would be

  • \(400\,000 \times 1.03^{10} \approx\) 537 567 dollars. "If" is the key word: prices may rise more slowly, or fall.

6. A bank account. Account X costs 4 dollars a month and includes 12 transactions. Account Y has no monthly fee but charges 1.25 dollars per transaction. You make 10 transactions a month.

  • X: 4 dollars. Y: \(10 \times 1.25 = 12.50\) dollars. X is cheaper for you. (Break-even: \(4 \div 1.25 = 3.2\) transactions; above 3 a month, X wins.)

Try it

  1. Gym A costs 30 dollars a month. Gym B charges 10 dollars a visit. How many visits a month make them cost the same? Which should someone who goes twice a week choose?
  2. A streaming service costs 12.99 dollars a month, or 119.99 dollars for a year paid at once. How much does the yearly option save? What percent is that?
  3. A 24 000 dollar car loses 20% of its value each year. What is it worth after 3 years?
  4. A 1500 dollar laptop loses 300 dollars of value each year (straight-line). What is it worth after 3 years? What would it be worth if it lost 30% a year instead?
  5. Lease: 1500 dollars up front and 350 dollars a month for 36 months. Buy: 28 000 dollars, and sell for 16 000 dollars after 3 years. Which is cheaper over 3 years, and by how much (ignoring interest and other costs)?
  6. A card offers "0% for 6 months, then 24% a year". You buy a 1200 dollar sofa and pay 100 dollars a month for 6 months. Then interest starts on what is left. What is the first month's interest after the 6 months?
  7. If a 350 000 dollar home appreciated by 4% a year for 5 years, what would it be worth?
  8. Plan A costs 60 dollars a month with plenty of data. Plan B costs 40 dollars plus 2 dollars per GB. Find the break-even point.
Answers
  1. \(30 \div 10 = 3\) visits. Twice a week is about 8 or 9 visits a month, which would cost 80 or 90 dollars at Gym B, so Gym A is far cheaper for them.
  2. Monthly for a year: \(12 \times 12.99 = 155.88\) dollars. Saving: \(155.88 - 119.99 = 35.89\) dollars, which is \(\dfrac{35.89}{155.88} \approx 23\%\).
  3. \(24\,000 \times 0.80^3 = 24\,000 \times 0.512 = 12\,288\) dollars.
  4. Straight-line: \(1500 - 3 \times 300 = 600\) dollars. At 30% a year: \(1500 \times 0.70^3 = 1500 \times 0.343 = 514.50\) dollars.
  5. Lease: \(1500 + 36 \times 350 = 14\,100\) dollars. Buy: \(28\,000 - 16\,000 = 12\,000\) dollars. Buying is cheaper by 2100 dollars in this example.
  6. After 6 payments, \(1200 - 600 = 600\) dollars is left. One month at 24% a year: \(600 \times \dfrac{0.24}{12} = 12\) dollars.
  7. \(350\,000 \times 1.04^5 \approx 425\,829\) dollars (if prices really did rise that way).
  8. \(40 + 2g = 60\), so \(g = 10\) GB. Under 10 GB, Plan B is cheaper; over 10 GB, Plan A is.

Common mistake

Comparing monthly prices instead of totals. A lower monthly payment can cost more overall, because it lasts longer or hides the price of something else. Always multiply out to the same period.

Forgetting what something is worth at the end. A bought car still has value after 4 years; a leased one does not belong to you. A fair comparison counts that.

Where it leads